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Control constrained optimal control problems in non-convex three dimensional polyhedral domains

By Gunter Winkler

Abstract

The work selects a specific issue from the numerical analysis of optimal control problems. We investigate a linear-quadratic optimal control problem based on a partial differential equation on 3-dimensional non-convex domains. Based on efficient solution methods for the partial differential equation an algorithm known from control theory is applied. Now the main objectives are to prove that there is no degradation in efficiency and to verify the result by numerical experiments. We describe a solution method which has second order convergence, although the intermediate control approximations are piecewise constant functions. This superconvergence property is gained from a special projection operator which generates a piecewise constant approximation that has a supercloseness property, from a sufficiently graded mesh which compensates the singularities introduced by the non-convex domain, and from a discretization condition which eliminates some pathological cases. Both isotropic and anisotropic discretizations are investigated and similar superconvergence properties are proven. A model problem is presented and important results from the regularity theory of solutions to partial differential equation in non-convex domains have been collected in the first chapters. Then a collection of statements from the finite element analysis and corresponding numerical solution strategies is given. Here we show newly developed tools regarding error estimates and projections into finite element spaces. These tools are necessary to achieve the main results. Known fundamental statements from control theory are applied to the given model problems and certain conditions on the discretization are defined. Then we describe the implementation used to solve the model problems and present all computed results

Topics: info:eu-repo/classification/ddc/510, ddc:510, Anisotropes Gitter, Elliptische Differentialgleichung, Finite-Elemente-Methode, Optimale Kontrolle, anisotropic finite elements, linear-quadratic optimal control problem, non-convex domain, superconvergence
Year: 2008
OAI identifier: oai:qucosa:de:qucosa:18908
Provided by: Qucosa
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